Confidence Interval

The confidence interval gives a range centered around the sample mean () and it indicates how closely we believe our sample mean is representing the population mean (greek letter mu). A large confidence interval suggests that the sample does not provide a precise representation of the population mean, whereas a narrow confidence interval demonstrates a greater degree of precision.

The formula used to calculate the confidence interval of a sample is

Confidence Interval = Sample Mean (plus or minus) the t-value times s sub x-bar

All confidence intervals are expressed according to a particular probability (also referred to as level of confidence) that the interval correctly includes the population mean. This caveat is necessary because the population parameters remain unknown, but are estimated by theoretical inferences from the sample statistics. Therefore, confidence intervals become wider as we want to be more certain that the population mean is included. The probability level is incorporated into the equation by t or the t-value. Probability levels conventionally assigned to confidence intervals included 80%, 90% or 95%, and can be obtained from t-tables presented in most statistics textbooks. A t-value of 1.96 is used for a 95% confidence interval for normally distributed samples with a sample size greater than 30.

The other factor influencing confidence interval width is the standard error (s sub x-bar), which indicates variability of the population and the degree to which it is mediated by sample size.

Confidence intervals are used in statistical analysis to describe the probability that two sample means are from the same population, provided that the data sets exhibit a normal distribution. If there is overlap between the confidence intervals for two means being compared, then it is concluded that these values are not significantly different with a probability equivalent to the probability used to establish the t-value.

References and Further Reading

Bonham, C.D. 1989. Measurements for terrestrial vegetation. John Wiley Son, New York, NY. pp 64-65.

Cook, C.W., and J. Stubbendieck. (eds). 1986. Range research: Basic problems and techniques. Society for Range Management, Denver, CO. pp 216-219.